Question

So sánh :

a, \(\sqrt{11}\)+\(\sqrt{13}\) và 2. \(\sqrt{12}\)

b, \(\sqrt{69}\)-\(\sqrt{68}\)và \(\sqrt{68}\)-\(\sqrt{67}\)

Answer 1

a) Ta có : \(\left(\sqrt{11}+\sqrt{13}\right)^2=11+2\sqrt{11.13}+13=24+2\sqrt{143}\)

\(\left(2.\sqrt{12}\right)^2=4.12=24+2.\sqrt{144}\)

mà \(\sqrt{144}>\sqrt{143}\Rightarrow24+2\sqrt{144}>24+2\sqrt{143}\Rightarrow\left(2.\sqrt{12}\right)^2>\left(\sqrt{11}+\sqrt{13}\right)^2\)

\(2.\sqrt{12}>\sqrt{11}+\sqrt{13}\)

Answer 2

b) Ta có : \(\left(\sqrt{69}-\sqrt{68}\right)-\left(\sqrt{68}-\sqrt{69}\right)\)

        \(\Leftrightarrow\sqrt{69}+\sqrt{67}-2\sqrt{68}\)

Từ kq câu a \(\Rightarrow\sqrt{69}+\sqrt{67}< 2\sqrt{68}\)

\(\Rightarrow\sqrt{69}+\sqrt{67}-2\sqrt{68}< 0\)

\(\Rightarrow\left(\sqrt{69}-\sqrt{68}\right)-\left(\sqrt{68}-\sqrt{67}\right)< 0\)

\(\Rightarrow\sqrt{69}-\sqrt{68}< \sqrt{68}-\sqrt{67}\)